Sunflower vs non-sunflower clique behavior in type A when k is proportional to ℓ

Determine the precise behavior of sunflower and non-sunflower cliques in the graphs Γ(A_ℓ, k) when k is proportional to ℓ. Specifically, characterize how clique structure varies in this regime for Γ(A_ℓ, k), where vertices are sums of k-element strongly orthogonal subsets of A_ℓ roots and edges connect pairs whose difference is also such a sum.

Background

Theorem 4.1 (from prior work) shows that for A_ℓ and ℓ > k·4k, every maximal clique in Γ(A_ℓ, k) is a sunflower. The exceptional types studied here show low sunflower proportions, suggesting the large-gap condition may be necessary.

The authors explicitly note that understanding the regime k ≍ ℓ is unresolved, asking for the precise behavior of sunflower and non-sunflower cliques in Γ(A_ℓ, k) when k is proportional to ℓ.

References

This suggests that the condition $\ell \gg k$ in Theorem \ref{BGOmain} is likely to be necessary; it remains open to determine the precise behaviour of sunflower and non-sunflower cliques in the $A_\ell$ system when $k$ is proportional to $\ell$.

— Cliques in graphs constructed from Strongly Orthogonal Subsets in exceptional root systems  (2604.02983 - Browne et al., 3 Apr 2026) in Section 5 (Open Questions and Concluding Remarks)